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\DOI{10.5802/crgeos.224}
\datereceived{2023-01-24}
\daterevised{2023-05-09}
\dateaccepted{2023-06-08}
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\dateposted{2023-06-27}
\begin{document}

\begin{noXML}

\RubricEF{Tectonics, tectonophysics, geodynamics}{Tectonique, tectonophysique, g\'eodynamique}

\TopicEF{Research article}{Article de recherche}

\shortrunauthors

\title{Influence of multiple d\'{e}collement and cover rock rheology on
the structural evolution of thin-skinned fold-and-thrust belts:
insights from discrete element modelling}

\author{\firstname{Nazanin} \lastname{Eslamirezaei}\CDRorcid{0000-0002-5380-3160}\IsCorresp}
\address{Faculty of Earth Sciences, Department of Geology, 
Shahid Beheshti University, Tehran, Iran} 
\email[N. Eslamirezaei]{nazanin.eslami@gmail.com}

\author{\firstname{Seyed Ahmad} \lastname{Alavi}\CDRorcid{0000-0002-2393-9526}}
\addressSameAs{1}{Faculty of Earth Sciences, 
Department of Geology, Shahid Beheshti University, Tehran, Iran} 
\email[S. A. Alavi]{a-alavi@sbu.ac.ir}

\author{\firstname{Seyed Tohid} \lastname{Nabavi}\CDRorcid{0000-0003-1163-3850}}
\addressSameAs{1}{Faculty of Earth Sciences, 
Department of Geology, Shahid Beheshti University, Tehran, Iran} 
\email[S. T. Nabavi]{tohidnabavi@gmail.com}

\author{\firstname{Mohammad Reza} \lastname{Ghassemi}\CDRorcid{0000-0002-0641-0073}}
\address{Research Institute for Earth Sciences, 
Geological Survey of Iran, Tehran, Iran}
\email[M. R. Ghassemi]{ghassemimr@gmail.com}

\keywords{\kwd{Structural style}
\kwd{Mechanical evolution}
\kwd{Multiple d\'{e}collements}
\kwd{Thin-skinned tectonics}
\kwd{Fold-and-thrust belt}
\kwd{Discrete element model}}

\begin{abstract} 
A variety of thin-skinned fold-and-thrust belts are associated with
multiple d\'{e}collements, which have low frictional basal and
intermediate d\'{e}collements related to mechanical stratigraphy and or
overpressure conditions. The present study considers six ternary series
of numerical discrete element (DE) models to simulate and explore the
effects of mechanical stratigraphy with varied d\'{e}collement layers,
the number of cover sequences and thickness in the structural style,
evolution, and strain partitioning of thin-skinned fold-and-thrust
belts. Horizontal layer-parallel shortening of particles was induced by
horizontal motion of a vertical boundary wall. The modelling results
show weak d\'{e}collements promoted the decoupled deformation of the
fold-and-thrust belt with continues shortening. The results indicate
that shortening was mainly accommodating by thrust-related folds with
significant differences in the structural style. The formation of
d\'{e}collement, box, fault-propagation, and fault-bend folds, as well
as extensional faults represents the ultimate style of the
fold-and-thrust belts developed in DE-models. DE-models with thin and
fewer number of d\'{e}collements demonstrate that box-shaped
d\'{e}collement folds with less secondary disharmonic folds developed
in the limbs of main fold structures. Furthermore, DE-models with
thickest upper d\'{e}collements are characterized by structural
decoupling, more flow of material into the core of anticlinal
structures and formation of complicated structures. The results of this
study show that the structural style and decoupling can be affected by
the rheology, number, and thickness of d\'{e}collements. The
DE-modelling results compared with natural examples, as well as
analogue and numerical models, show that our mechanical modelling can
overall match thin-skinned fold-and-thrust belts with multiple
d\'{e}collements that present different structural style.
{\vspace*{-2pt}}
\end{abstract}

\maketitle

%\vspace*{3pc plus 1pt minus 1pt}

\twocolumngrid

\end{noXML}

\section{Introduction}\label{sec1} 
Field-based observations, seismic reflection profiles, balanced and
restored crustal-scale sections, as well as analogue and numerical
models demonstrated the geometry and kinematics observed in
contractional settings such as fold-and-thrust belts are controlled by
various factors. These factors have been studied, such as the presence
of pre-existing structures 
\citep[e.g.,][]{Nilforoushanetal2013,Nabavietal2020,DelVentisetteetal2021,Najafietal2021,Shamszadehetal2022a},
basement morphology and thrust ramp dip angle 
\citep[e.g.,][]{MaillotKoyi2006,Rosasetal2017,Caeretal2018,Ghoshetal2020},
the depositional environment, mechanical stratigraphy (i.e. the
mechanical properties, thickness of layers, and interface properties of
rock units)  \citep[e.g.,][]{Cruzetal2008,Farzipour-SaeinKoyi2014,Plaetal2019,ItoMoore2021}, the presence of d\'{e}collements
(single and/or multiple) and their initial configurations 
\citep[e.g.,][]{Sherkatietal2006,Stockmaletal2007,Graveleauetal2012,
Nilforoushanetal2012,Ruhetal2012a,Ruhetal2013a,Ruhetal2017,Watkinsetal2014,
Ghanadianetal2017a,Ghanadianetal2017b,LiMitra2017,Borderieetal2018,
Derikvandetal2018,Derikvandetal2019,Heydarzadehetal2020,
Zhidongetal2021, Eslamirezaeietal2022,Mohammadrezaeietal2022}, 
the basal slope and velocity 
\citep[e.g.,][]{Rosasetal2017}, the basal friction 
\citep[e.g.,][]{CottonKoyi2000,CostaVendeville2002,KoyiMaillot2007},
the angle of convergence 
\citep[e.g.,][]{Casasetal2001,McClayetal2004,Leeveretal2011,Barcosetal2016,Nabavietal2017a,Nabavietal2017b,Nabavietal2020},
strain rate 
\citep[e.g.,][]{Smitetal2003,Ruhetal2014a},
and syn-kinematic
(syn-tectonic) surface processes (i.e. sedimentation and erosion) 
\citep[e.g.,][]{PichotNalpas2009,Cruzetal2010,Cruzetal2019,WuMcClay2011,Driehausetal2014,Derikvandetal2019,Plaetal2019,DalZilioetal2020,Xuetal2021}.
A class of fault-related folds is d\'{e}collement folds, which are
folding of more competent layers above a weak d\'{e}collement without a
ramp fault coring the uplift with a decreasing in displacement on a
d\'{e}collement  \citep{Mitra2003,Fossen2016,NabaviFossen2021}. 
A variety of thin-skinned fold-and-thrust belts (i.e.
d\'{e}collement-dominated interpretive structural style and where
deformation restricted to the sedimentary cover above the
d\'{e}collement level) are developed associated with multiple
d\'{e}collements such as the Zagros fold-and-thrust belts 
\citep[e.g.,][]{Sepehretal2004,
Sherkatietal2006,Vergesetal2011,Yosefnejadetal2017,Derikvandetal2018,MengHodgetts2019a,Shamszadehetal2022b},
the Jura Mountains
\citep[e.g.,][]{Schorietal2015}, the Pyrenees 
\citep[e.g.,][]{KoyiSans2003}, 
which exhibit low basal, intermediate
and even upper friction related to lithology (salt, shale, marl) and/or
overpressure\unskip\break 
conditions.

Many studies have already recognized that the density and thickness of
d\'{e}collements have played a significant role in the structural
evolution and style, strain distribution and partitioning, controlling
coupling and decoupling processes, and syn-kinematic processes in
thin-skinned fold-and-thrust belts 
\citep[e.g.,][]{CottonKoyi2000,Koyietal2000,Mitra2003,Sherkatietal2005,
Vergesetal2011,Ruhetal2012a,Ruhetal2013a,Schorietal2015,MengHodgetts2019a,
MengHodgetts2019b,Zhidongetal2021,NabaviFossen2021,Sunetal2021,Xuetal2021}. 
Low-friction d\'{e}collement is much weaker than any other layer in the
rock sequences, so allowing material to flow at geologically high
strain rates in response to normal and shear stresses of around 1~MPa.
Moreover, portions dominated by folding are commonly underlain by a
low-friction d\'{e}collement and the flow of weak material into the
cores of anticlinal structures may be inevitable for the evolution of
d\'{e}collement folds  
\citep[e.g.,][]{Burberryetal2010,
Cuietal2019}. 

Although previous studies have achieved an understanding of the role of
d\'{e}collements in the structural evolution of thin-skinned
fold-and-thrust belts. This issue remains questionable, and numerical
models provide an understanding of deformation patterns and structural
styles. Proper interpretation of the structural evolution and
structural style are essential to build the accurate understanding of
the surface and subsurface geometry of structures. In this way,
numerical modelling using discrete element method (DEM), as a
particle-based method, provides means for simulating and tracking the
spatial and temporal evolution of stress distribution, strain
localization and sequential evolution of structures. It has been
recognized that fold-and-thrust belts evolved in
d\'{e}collement-dominated way and exhibit wide range of structural
styles, different fault-fold interactions, and associated fault-related
folds, the mechanical conditions that contribute to these variations
have not well to be understood.

This study makes use of a series of discrete element numerical models
using the Itasca Particle Flow Code 
(PFC3D\tsup{\text{\textregistered}}, version 6,13, Itasca Consulting
Group) to investigate the formation, development and evolution of
thin-skinned fold-and-thrust belt comprising pre-kinematic
low-frictional d\'{e}collements , the number of d\'{e}collements,
thickness of d\'{e}collements, the sequence of competent and less
competent layers within the sedimentary cover sequence under an applied
horizontal pressure and the amount of shortening. In this context, the
present study aims to understand how the presence of low-friction
multiple d\'{e}collements influence the deformation style of
thin-skinned fold-and-thrust belts. These mechanical properties have a
direct influence on the deformation pattern of individual layers as
well as the entire stratigraphic sequence.

Geological processes such as diagenesis, erosion, fluid flow,
temperature, etc. are not considered. The primary reason for this
approach is that we seek to isolate the effects of a few selected
parameters in order to keep the analysis traceable. Besides, the
numerical models can provide a powerful tool both to help interpret the
structural style affected by d\'{e}collements that may be poorly imaged
on seismic profiles and previous models and to understand the
kinematics and mechanics of fold-and-thrust belts which include more
than a d\'{e}collement. Finally, the modelling results of the present
study are compared with natural examples, as well as analogue and
numerical models.

\section{The discrete element modelling method (DEM)}\label{sec2} 

\subsection{Basic principles}\label{sec2.1} 
To develop an accurate understanding of the geometry of structures, the
detailed analysis of structural evolution and structural style is
needed. Numerical modelling using the Discrete Element Method (DEM) can
simulate and monitor the spatial and temporal evolution of stress
distribution as a particle-based approach. DEM, in except of
time-consuming calibration of particle parameters to material
properties, is an effective method to investigate the initiation,
propagation, interaction and linkage, and evolution of folding,
fracturing, and faulting in a practical manner involving deformation
and large relative displacements.\pagebreak
Many previous researches have used
the DE-models and have been successfully produced realistic styles of
geological and tectonic features such as normal faulting in layered
sequences  \citep[e.g.,][]{Botteretal2014,Hardy2019}, d\'{e}collement
folding  
\citep[e.g.,][]{HardyFinch2005,Eslamirezaeietal2022},
fault-bend folding  \citep[e.g.,][]{Beneshetal2007,Hughesetal2014},
fault-propagation folding
\citep[e.g.,][]{HardyFinch2006,HughesShaw2015}, and fold-and-thrust
belts 
\citep[e.g.,][]{BurbidgeBraun2002,Hardyetal2009,Morgan2015,MengHodgetts2019a,MengHodgetts2019b}. 

A series of DE-mechanical models, are applied for simulation of
deformation of the rock layers in a shortening setting. In this
approach that uses a time stepping, finite difference approach to solve
Newton's equations of motion, the rock mass is assumed to be composed
of soft-circular, disk, or disc particles that interact in pairs as if
bound by breakable elastic bonds (springs) and move relative to one
another. These particles interact with elastic, gravitational,
frictional and viscous forces under applied forces and displacement
boundary (or wall) conditions. At each discrete time step, the
particles are displaced to their new positions within the model by
integrating their equations of motion using Newtonian laws and a
velocity-Verlet-based numerical scheme, which is a numerical method
used to integrate Newton's equations of motion as this integrator
strike a good balance between computational cost, stability, and
accuracy  \citep{Verlet1967}. The particle bonding is controlled by
normal and shear bond stiffness, bond strength, and bond radius. In
this regard, the normal and shear contact forces (Mohr--Coulomb
criterion) are calculated between two overlapping or contacting
particles using a rheological contact law. The bonds can be broken when
the inter-particle forces exceed the bond strength. In this respect,
slip can occur between particles. Thus, discontinuities and
heterogeneities can form and evolve in response to changing stress
condition and material properties 
\citep[e.g.,][]{CundallStrack1979,Hardyetal2009,Hughesetal2014,Morgan2015,Smartetal2011,Hardy2019,MengHodgetts2019a,MengHodgetts2019b,Eslamirezaeietal2022}. 

\begin{figure*}
{\vspace*{3pt}}
\includegraphics{fig01}
{\vspace*{3pt}}
\caption{\label{fig1}Initial design and boundary condition of discrete
element models. Models are consisted of a basal d\'{e}collement and
adjacent layers of less competent and competent layers; middle and
upper d\'{e}collements will have added in the process of modelling. The
$f_{s} = \mbox{shear stress}$, $f_{n} = \mbox{normal stress}$ and $b$
is particle bond.}
{\vspace*{3pt}}
\end{figure*}

\subsection{Model setup}\label{sec2.2}  
In the present study, we present a series of particle-based DE-models
using the Itasca Particle Flow Code, named  
PFC3D\tsup{\text{\textregistered}}\ software  package (vers3ion 6.13,
Itasca Consulting Group, 
\href{https://www.itascacg.com/software/pfc}{https://www.itascacg.com/\break{}software/pfc}). 
PFC\tsup{\text{\textregistered}}\ was initially developed for
describing mechanical behavior of material (soil or rocks) and allows
much flexibility in simulations and is widely used among geologists as
it allows to model large deformations. We created a regular initial
model  (Figure~\ref{fig1}) consisting of 21403 spherical particles in
the part of upper crust model with dimensions of 10 km length
${\times}$ 1~km width ${\times}$ 0.5~km thickness $(x,y,z)$. The
particles radii range from 25 to 41~m for both d\'{e}collements and
cover layers following the Gaussian distribution of particle size. The
mechanically layered cover sequence consists 
of\unskip\break 
relatively competent
layer and relatively less competent layers. The average rock density
for all particles is about  3000 kg/m$^{3}$. Particle stiffness (both
shear and normal) were assigned with values of $1 \times 10^{9}$,  $0.5
\times 10^{9}$, and  $0.1 \times 10^{9}$~N/m, respectively, for
competent, less competent, and weak d\'{e}collements,
(Table~\ref{tab1}). Inter-particle friction (${\upmu}$) was assigned as
0.5 for cover layers, whereas 0 for d\'{e}collements that are
frictionless to ensure their low strength 
\citep[e.g.,][]{Morgan2015,MengHodgetts2019a}. All over, we examined
different models based on changing variations included d\'{e}collement
number and thickness and the position and number of competent and less
competent layers within the sedimentary cover. Colored layered within
the cover sequence above and adjacent to d\'{e}collements serve as
strain markers during progressive deformation of the models and
indicate contrast in material properties.

\begin{table*}
\caption{\label{tab1}Summary of material properties and parameters used
in 3D DE-models\vspace*{-2pt}}
\begin{tabular}{lccc}
\thead
\xmorerows{1}{Parameters} & 
\xmorerows{1}{D\'{e}collements layers} &
\multicolumn{2}{c}{Overburden}\\
\cline{3-4}
 & & Less competent layers & Competent layers\\
\endthead
Particle radius (m) & \multicolumn{3}{c}{25--41~m}\\
Particle Stiffness (N/m) & $0.1\times10^9$ & $0.5\times10^9$ & $1\times10^9$\\
Contact friction $(\upmu)$ & 0 & \multicolumn{2}{c}{0.5}\\ 
Tensile bond strength (N) & 0 & $2.5\times10^6$ &$5\times10^6$\\
Average density (kg/m$^3$) &\multicolumn{3}{c}{3000~kg/m$^3$}
\botline
\end{tabular}
\vspace*{-2pt}
\end{table*}

Our experimental setup consists of a horizontal surface and four
vertical model boundaries. All of these surfaces are defined in the
models by rigid, frictional boundary walls (i.e. free-slip boundary
conditions). Numerically, free surface boundary conditions are applied
on the top model face, thus permitting self-consistent evolution of
topography  \citep{Nabavietal2020}. Acceleration due to gravity 
(9.81~m/s$^{2}$) is imposed throughout the deformation as a body force.
In all study models, a horizontal, vertical, compressional load was
imposed along the left sidewall  [i.e. displacement-based boundary
conditions;  \citealp{Nabavietal2018,Nabavietal2020}]. All results are
shown here after 10\%, 20\%, and 30\% shortening which are,
respectively, equal to  1000~m, 2000~m and 3000~m displacement on
d\'{e}collements. The shear strength is  $5 \times 10^{6}$ and  $2.5
\times 10^{6}$ for competent and less competent layers respectively.

\section{Modelling results}\label{sec3}
We present here the main features observed in the six series of
DE-models after  30\% (${\sim}$3000~m displacement) orthogonal
shortening to investigate the deformation and structural evolution of
thin-skinned fold-and-thrust belts with multiple d\'{e}collements
(summarized in  Table~\ref{tab2}). Clearly, there are large number of
d\'{e}collements model setups that we could present, however, we
discussed here: (1)~the role of multiple d\'{e}collements, 
(2)~d\'{e}collement thickness, (3)~the number and mechanical properties
of competent and less competent layers within the sedimentary cover,
and (4)~strain localization in the evolution of thin-skinned
fold-and-thrust belts and their structural style using six series of
DE-models which are based on progressive deformation events. In all
models, the sedimentary cover sequence overly a d\'{e}collement at the
base of models. Furthermore, we consider more units as d\'{e}collements
to simulate multiple d\'{e}collements models. Development and the
evolution of deformation in a series of DE-models illustrate that
displacement field vary in different parts of the fold and also a
fold-and-thrust belts and they play an important role in fold
geometries and the structural style of a contractional tectonic
setting. 

\vspace*{-1pt}

\subsection{Models with two d\'{e}collements}\label{sec3.1} 

\vspace*{-1pt}

\subsubsection{Series 1---an upper d\'{e}collement surrounded by
competent layers}\label{sec3.1.1} 
Model 1 (Figure~\ref{fig2}) presents the effects of basal
d\'{e}collement and a thick intermediate d\'{e}collement on the
structural style and evolution of the sedimentary cover sequence
consisted of a series of competent bonded layers (four competent layers
thick) intercalated with less competent bonded layers (three less
competent layers thick). Overall, after an \mbox{initial} step of homogeneous
shortening of the model, the deformation is essentially pure-shear
thickening (contraction or buckling) with the hinterland-verging
low-amplitude asymmetric d\'{e}collement fold at the onset of
deformation. In this model, the first contractional structures such as
asymmetric d\'{e}collement folds and some fore-thrusts appeared after
20\% bulk shortening. Also, completely detached rootless folds appeared
above the intermediate d\'{e}collement (Model~1c in 
Figure~\ref{fig2}). After 30\% of bulk shortening, the model shows
disharmonic geometry of deformation within the layers due to the
difference in propagation between the layers above and beneath the
d\'{e}collements from surface to deeper parts. Shortening led to
development of more penetrative contractional structures such as
back-thrusts, faulted d\'{e}collement fold, and hinterland-verging
fault-propagation fold (Model 1d in  Figure~\ref{fig2}). Also, some
detached rootless folds above the intermediate d\'{e}collement were
still visible. Moreover, increasing shortening causes both flexural
slip and flexural flow, respectively, in major shear zones and in the
lower parts of fault-related folds so that they are filled with less
competent material that evacuated beneath synclines and flowed into the
anticline core.

\begin{figure*}
\vspace*{3pt}
\includegraphics{fig02}
\vspace*{5pt}
\caption{\label{fig2}The velocity vectors for sedimentary sequence and
the evolution of DEM to show the folding in the models 1, 2, and 3 of
series 1 for sedimentary sequence with two d\'{e}collements after 30\%
(3000~m displacement) orthogonal shortening in three steps of
shortening 10\%, 20\%, and 30\%. The models are the same in terms of
the thickness of the basal d\'{e}collement and different in the upper
d\'{e}collement with increasing thickness surrounded by competent
layers.}
\vspace*{14pt}
\end{figure*}

\begin{table*}
\caption{\label{tab2}Summary of results of the discrete element models}
\tabcolsep=2pt
\fontsize{7.75}{9}\selectfont
\begin{tabular}{lccccccccccccccccccc}
\thead
\xmorerows{1}{Model name} &
\multicolumn{3}{c}{Series 1} &
\multicolumn{3}{c}{Series 2} &
\multicolumn{3}{c}{Series 3} & 
\multicolumn{3}{c}{Series 4} &
\multicolumn{3}{c}{Series 5} &
\multicolumn{3}{c}{Series 6}
\\\cline{2-4}\cline{5-7}\cline{8-10}\cline{11-13}\cline{14-16}\cline{17-19}
 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 & 1 & 2 & 3 \\
\endthead
\parbox[t]{3.8pc}{\raggedright\hangindent .3pc
Number of \mbox{d\'{e}collements}} & 
2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 
\vspace*{6pt}\\ 

\morerows{1}{\parbox[t]{3pc}{\raggedright\hangindent .6pc
Main fold styles}} & \morerows{1}{BF} & FPF & BF &
\morerows{1}{FBF} &  \morerows{1}{FBF} & \morerows{1}{FBF} &
\morerows{1}{BF} & \morerows{1}{BF} & \morerows{1}{BF} &
\morerows{1}{BF} &  \morerows{1}{BF} & \morerows{1}{DF} &
\morerows{1}{BF} & \morerows{1}{BF} & BF & \morerows{1}{BF} & 
\morerows{1}{FPF} & DF \\ 
 & & FBF & FPF & & & & & & & & & & & & DF & & & BF
\vspace*{6pt} \\ 

\morerows{1}{\parbox[t]{3pc}{\raggedright\hangindent .6pc D\'{e}collement thickest}} &
\morerows{1}{-} & \morerows{1}{-} & \morerows{1}{Intermediate} & 
\morerows{1}{-} & \morerows{1}{-} & \morerows{1}{Intermediate} & 
\morerows{1}{-} & \morerows{1}{-} & \morerows{1}{Upper} & \morerows{1}{-} &
\morerows{1}{-} & Intermediate& \morerows{1}{-} & \morerows{1}{-} & 
\morerows{1}{Upper} & \morerows{1}{-} & \morerows{1}{-} & 
\morerows{1}{Intermediate} \\
&&&&&&&&&&&&Upper\vspace*{2pt} &&&&&& 
\botline
\end{tabular}
\tabnote{BF $=$ Box Fold. FBF $=$ Fold Bend Fold. 
FPF $=$ Fold Propagation Fold. DF $=$ D\'{e}collement fold.}
\vspace*{4pt}
\end{table*}

Model 2  (Figure~\ref{fig2}) is dominated by faulting, however, the dip
angle of faults varies as shortening progresses and also in accordance
with mechanical stratigraphy (composition). In this regard, the low
angle or sole thrust fault was developed in the lower d\'{e}collement
and propagate upward within the overlying competent sediments so that
faults passing through competent units are steeper than those within
incompetent units (i.e. a faulted d\'{e}collement fold structure).
Hence, as deformation continued, the fault progressively elongated
along basal and upper d\'{e}collements and such variations in
mechanical stratigraphy of the deformation setting leads to the
development of ramp--flat geometries and associated structures such as
fault-bend folds, so that only one fault broke to the lower unit and
thrust it up over the upper unit. Moreover, as deformation continued,
the fault as a back-thrust progressively elongated until a second fault
formed in front and propagated from the upper d\'{e}collement to the
surface, and also from the foreland to the hinterland. Shortening led
to formation and development of more penetrative contractional
structures such as backthrusts, faulted d\'{e}collement fold. Also,
some detached rootless folds and blind thrusts above the 
intermediate\unskip\break 
d\'{e}collement were still visible. Total (relative) displacement
magnitude and direction between the driving hinterland wall and the
foreland wall are illustrated and colored as the characteristic of the
tectonic shortening accommodated by internal deformation of the model
(Model 2e in  Figure~\ref{fig2}). The primary control on throw
partitioning in the model is fault displacement. The displacement
magnitude field is smooth, fan-like and radiates from the basement
fault tip. At low differential displacements, undeformed state to
gentle folding of beds is the main contributor to deformation
accommodated across the contractional zone. Backthrusts occur also in
the model and show relatively low displacement, so that slip is
predominantly along
forethrusts.

Model 3  (Figure~\ref{fig2}) shows a simulation of sedimentary layers
with intermediate d\'{e}collement, which affect the structural style
and evolution of the sedimentary cover sequence consisted of a series
of competent bonded layers (four competent layers thick) intercalated
with less competent bonded layers (two less competent layers thick).
After 20\% bulk shortening, the lower structural level of the study
model initially forms a lift-off detachment fold, and subsequently
became a hinterland-verging asymmetric d\'{e}collement anticlinal
structures as a backthrust to accommodate the deformation. The upper
level shows a train of buckle folds without any geometric connection to
the lower part. After 30\% bulk shortening, asymmetric d\'{e}collement
folds in the lower structural level of the study model evolve into
hinterland-verging fault-propagation folds cored by the basal
d\'{e}collement material (i.e. a faulted d\'{e}collement fold
structure). Furthermore, shortening was accommodated by the formation
of new buckle folds and also amplification and tightening of existing
folds in the upper level (Model 3d in  Figure~\ref{fig2}). Underlying
folds vergence in the same direction as the 
overlying\unskip\break 
structures, suggesting that deformation has been only partially
decoupled across the intermediate d\'{e}collement.

\subsubsection{Series 2---an upper d\'{e}collement surrounded by
less-competent layers}\label{sec3.1.2}
In Series 2  (Figure~\ref{fig3}) we investigate the effect of basal and
upper d\'{e}collements and the upper d\'{e}collement variation in
thickness on structural style and evolution of sedimentary cover
sequences consisted of a series of less competent bonded layers (four
less competent layers thick) intercalated with competent bonded layers
(two competent layers thick). Overall, after an initial step of
homogeneous shortening of model 1 of series 2 experiments, deformation
initiates by pure-shear thickening that caused formation of two
low-amplitude asymmetric folds with disharmonic minor folds and
thickening in both limbs near hinterland and un-deformation in foreland
at the onset of deformation. In this model, the first contractional
structures such as low amplitude asymmetric folds and some disharmonic
folds appeared after 20\% of shortening (Model 1c 
in\unskip\break  
Figure~\ref{fig3}). Furthermore, the folding accompanied by uplift and
martial flow toward the core of anticlines 
and formation of extensional
fracture/faults in the intermediate d\'{e}collement or forelimb of main
fold in final step after 30\% of shortening. At the end of experiment,
the model shows a general fault-bend style of folding, in which
shortening led to formation of detached rootless folds above
d\'{e}collement layers (Model 1d in  Figure~\ref{fig3}). In model 2
with thicker d\'{e}collement, the early stage of shortening and
deformation started by formation of foreland-verging fault-propagation
fold which was developed in the further stage by pure shear fault bend
fold and the height of the model increased as a result of contraction
in the 20\% of shortening (Model 2c in  Figure~\ref{fig3}). As bulk
shortening continues, fault segments were penetrated in the layers
beneath the upper d\'{e}collement and propagated upward by shortening
(Model~2c in  Figure~\ref{fig3}). In the 30\% of shortening the
thickening of upper d\'{e}collement appeared and deformation developed
by formation of asymmetric low amplitude fold with structural decupling
in final stage in 30\% (Model 2d in  Figure~\ref{fig3}).

\begin{figure*}
\includegraphics{fig03}
\caption{\label{fig3}The velocity vectors for sedimentary sequence and
the evolution of DEM to show the folding in the models 1, 2, and 3 of
series 2 for sedimentary sequence with two d\'{e}collements after 30\%
(3000~m displacement) orthogonal shortening in three steps of
shortening 10\%, 20\%, and 30\%. The models are the same in terms of
the thickness of the basal d\'{e}collement and different in the upper
d\'{e}collement with increasing thickness surrounded by less competent
layers.}
\vspace*{4pt}
\end{figure*}

In model 3 that was run to show a simulation of sedimentary layers
consisted of a series of less competent bonded layers (four less
competent layers thick) interacted with competent bonded layers (two
competent layers thick) with thickest intermediate d\'{e}collement in
this series  (Figure~\ref{fig3}).
Deformation\unskip\break    
initiates
by formation of
two main low amplitude folds in 20\% of shortening (Model 3c in 
Figure~\ref{fig3}) that developed to fault bend style in 30\% of
shortening (Model 3d in  Figure~\ref{fig3}). The thickening in the
upper d\'{e}collement was visible and structural decupling occurred by
formation of asymmetric folds in surficial layers. Furthermore,
disharmonic folds formed as a result of shortening at the final stage.
The shortening was accommodated by uplift and folding. Total (relative)
displacement magnitude and direction between the driving hinterland
wall and the foreland wall are illustrated and colored for all models
(Models 1e, 2e and 3e in  Figure~\ref{fig3}).

\subsubsection{Series 3---basal and upper d\'{e}collements separated
by sedimentary layers}\label{sec3.1.3}
In the series 3  (Figure~\ref{fig4}) we investigated the effects of
basal and upper d\'{e}collements and the variation in thickness of
upper one on the structural style and evolution of the sedimentary
cover sequence consisted of multiple competent and less competent
bonded layers that separated basal and upper d\'{e}collements. In model
1 of series 3 experiments the thickness of d\'{e}collements is equal to
other sedimentary layers. In the first stage of homogeneous shortening,
deformation initiates by flexural slip next to the hinterland that
produced disharmonic folds. Moreover, in the center of the model a
compressional box-shaped d\'{e}collement fold structure with two axial
surface and disharmonic folds in its limbs formed (Model 1c in 
Figure~\ref{fig4}). This box fold become tighten and developed by
uplift until 30\% of shortening (Model 1d in  Figure~\ref{fig4}).

\begin{figure*}
{\vspace*{-3pt}}
\includegraphics{fig04}
{\vspace*{-3pt}}
\caption{\label{fig4}The velocity vectors for sedimentary sequence and
the evolution of DEM to show the folding in the models 1, 2, and 3 of
series 3 for sedimentary sequence with two d\'{e}collements after 30\%
(3000~m displacement) orthogonal shortening in three steps of
shortening 10\%, 20\%, and 30\%. The models are the same in terms of
the thickness of the basal d\'{e}collement and different in the upper
d\'{e}collement with increasing thickness. Basal and upper
d\'{e}collements separated by several sedimentary layers.} 
{\vspace*{-4pt}}
\end{figure*}

In model 2 of series 3 experiments the upper d\'{e}collement is thicker
so deformation started by formation of symmetric d\'{e}collement folds.
In the 20\% of the shortening deformation continued by distortion in
the layers beneath the upper d\'{e}collement and thickening in the
d\'{e}collement layer (Model 2c in  Figure~\ref{fig4}). The layers
located on upper d\'{e}collement showed the low steep folding. Such
fold structures subsequently became a hinterland-verging
fault-propagation folds caused by the thickness and rheology of the
upper d\'{e}collement to accommodate the deformation (Model 2c in 
Figure~\ref{fig4}). The model 3 of series 3 experiments represents the
effect thickest upper d\'{e}collement on the structural style of
sedimentary sequence. In the 20\% of shortening, the asymmetric
box-shaped folds appeared that developed along with formation of minor
disharmonic folds (Model 3c in  Figure~\ref{fig4}). The shortening
accommodated by formation of disharmonic folds in the layers above the
upper d\'{e}collement. Shortening led to development of contractional
structures such as faulted d\'{e}collement fault, lift-off and
box-shaped folds above and beneath the upper d\'{e}collement (Model 3d
in  Figure~\ref{fig4}). Total displacement magnitude and direction
between the driving hinterland wall and the foreland wall are
illustrated and colored for all models as the characteristic of the
tectonic shortening accommodated by internal deformation of the model
(Models 1e, 2e and 3e in  Figure~\ref{fig4}).

{\vspace*{-2pt}}

\subsection{Models with three d\'{e}collements}\label{sec3.2} 

{\vspace*{-2pt}}

\subsubsection{Series 4---increasing thickness in intermediate and
upper d\'{e}collements}\label{sec3.2.1} 
The Series 4 (Figure~\ref{fig5}) presents the effect of multiple
d\'{e}collements on the structural style and evolution of the
sedimentary cover sequence consisted of a series of competent bonded
layers (six \mbox{competent} layers thick) intercalated with weak, less
competent bonded layers (three less competent layers thick). In this
series of models, the thickness of intermediate and upper
d\'{e}collements increased simultaneously. At the onset of deformation
applied on model 1 of series 4 experiments, shortening was accommodated
by development of a
series of box-shaped buckles in 20\% of shortening
(Model 1c in  Figure~\ref{fig5}) and the other box fold appeared in
next step along with uplift in 30\% of shortening and extensional
fracture/faults appeared in the final stage (Model 1d in 
Figure~\ref{fig5}). In model 2 of this series, the thickness of
intermediate and upper d\'{e}collement increased and deformation
started by formation of small wavelength asymmetric lift-off
d\'{e}collement fold with disharmonic minor folds and fractures in less
competent layers that developed further into box-shaped asymmetric fold
(Model 2c in  Figure~\ref{fig5}). Such fold structures subsequently
became a hinterland-verging fault-propagation folds caused by the
thickness and rheology of intermediate and upper d\'{e}collements to
accommodate the deformation (Model 2c in  Figure~\ref{fig5}). In the
next step of deformation, the layers are more tightened and other
contractional structures formed as presented in 30\% of bulk shortening
(Model 2d in  Figure~\ref{fig5}).

\begin{figure*}
\includegraphics{fig05}
\caption{\label{fig5}The velocity vectors for sedimentary sequence and
the evolution of DEM to show the folding in the models 1, 2, and 3 of
series 4 for sedimentary sequence with three d\'{e}collements after
30\% (3000~m displacement) orthogonal shortening in three steps of
shortening 10\%, 20\%, and 30\%. The models are the same in terms of
the thickness of the basal d\'{e}collement, intermediate and upper
d\'{e}collement with increasing thickness from 1 to 3.} 
\end{figure*}

The deformation in the Model 3 of Series 4 experiments started by
formation of symmetric d\'{e}collement fold that developed during
shortening by uplift and also extensional fracture/faults in the upper
d\'{e}collement. Martial fellow from d\'{e}collements into the core of
anticline is visible at the 30\% of shortening (Model 3d in 
Figure~\ref{fig5}). The type of d\'{e}collement folds in the final
stage is chevron lift-off fold whereby the layers are isoclinally
folded in the core of anticline and the ductile materials are flow from
the core of the folds. The shortening was accommodated by decoupling,
folding and uplift. Total displacement magnitude and direction between
the driving hinterland wall and the foreland wall are illustrated and
colored for all models (Models 1e, 2e and 3e in  Figure~\ref{fig5}) as
the characteristic of the tectonic shortening accommodated by internal
deformation of the model.

\subsubsection{Series 5---increasing thickness in the upper
d\'{e}collement}\label{sec3.2.2}
Series 5 (Figure~\ref{fig6}) presents to simulate the effects of three
d\'{e}collements on the structural style and evolution of the
sedimentary cover sequence consisted of a series of competent bonded
layers with the increasing thickness in the upper d\'{e}collement.
Overall, after an initial step of homogeneous shortening, the
deformation initiates by flexural slip in the model~1 of series 5
experiments that caused the formation of symmetric box-shaped folds and
disharmonic folds in the 20\% of shortening (Model 1c in 
Figure~\ref{fig6}) that evolved in the further stages of deformation by
uplift and also thickening in the sedimentary 
layers located in the
limbs of folds after 30\% of bulk\unskip\break 
shortening (Model 1d in 
Figure~\ref{fig6}). 
In model 2 of series 5 of experiments with thicker
upper d\'{e}collement, the deformation initiates by formation of
asymmetric folds in 20\% of shortening (Model 2c in 
Figure~\ref{fig6}). The folds progressively tighten and developed into
lift-off to box fold. This main lift-off fold amplified toward the
surface. The formation of forethrust in the core of box fold accrued in
30\% of shortening and extensional fracture/faults in the intermediate
d\'{e}collement presents the strain localization (Model 2d in 
Figure~\ref{fig6}). The shortening was accommodated by fold
amplification.

\begin{figure*}
{\vspace*{3pt}}
\includegraphics{fig06}
{\vspace*{3pt}}
\caption{\label{fig6}The velocity vectors for sedimentary sequence and
the evolution of DEM to show the folding in the models 1, 2, and 3 of
series 5 for sedimentary sequence with three d\'{e}collements after
30\% (3000~m displacement) orthogonal shortening in three steps of
shortening 10\%, 20\%, and 30\%. The models are the same in terms of
the thickness of basal and intermediate d\'{e}collements. The
thickness of the upper d\'{e}collement increases in the study models.}
{\vspace*{6pt}}
\end{figure*}

In model 3 of series 5 experiments with thickest upper d\'{e}collement
the deformation started by formation of low amplitude fold and an
asymmetric d\'{e}collement fold without any faults or fractures in 20\%
of shortening (Model 3d in  Figure~\ref{fig6}). By increasing the
shortening at the further steps in the 30\% of shortening, the highly
strain fold such as lift-off fold formed that contains the fractures in
the core of anticline (Model 3d in  Figure~\ref{fig6}). These fractures
in the layers beneath the upper d\'{e}collement are the results of
contractional environment and fold tightening. There are other
disharmonic folds in the layers near the surface that are effective in
the topography.\unskip\break    
Total (relative) displacement magnitude and direction
between the driving hinterland wall and the foreland wall are
illustrated and colored for all models (Models 1e, 2e and 3e in 
Figure~\ref{fig6}) as the characteristic of the tectonic shortening
accommodated by internal deformation of the model.

\subsubsection{Series 6---increasing thickness in the intermediate
d\'{e}collement}\label{sec3.2.3} 
In model series 6  (Figure~\ref{fig7}) that represents the effect of
multiple d\'{e}collements and the variation in thickness of
intermediate d\'{e}collement on the structural style and evolution of
the sedimentary cover sequence consisted of a series of competent
bonded layers (six competent layers thick) intercalated with less
competent bonded layers (three less competent layers thick).

\begin{figure*}
\includegraphics{fig07}
\caption{\label{fig7}The velocity vectors for sedimentary sequence and
the evolution of DEM to show the folding in the models 1, 2, and 3 of
series 6 for sedimentary sequence with three d\'{e}collements after
30\% (3000~m displacement) orthogonal shortening in three steps of
shortening 10\%, 20\%, and 30\%. The models are the same in terms of
the thickness of basal and upper d\'{e}collements. The thickness of
the intermediate d\'{e}collement increases in the study models. }
\end{figure*}

In model 1 of series 6 experiments, deformation initiates by uplift and
formation of single symmetric box-shaped d\'{e}collement folds and also
the \mbox{disharmonic} folds in sedimentary layers at the onset of deformation
(Model 1c in  Figure~\ref{fig7}). By increasing shortening, the other
box-shaped fold formed after 30\% of bulk shortening (Model 1d in 
Figure~\ref{fig7}). In model 2, minor fold structures developed
subsequently became a hinterland-verging fault-propagation folds after
20\% shortening
due to the thickness and rheology of intermediate
d\'{e}collement to accommodate the deformation (Model 2c in 
Figure~\ref{fig7}). After 30\% shortening, layers are isoclinally
folded in the core of the anticline and the fold developed by uplift
and formation of minor folds in the limbs of main folds and structural
decoupling and thickening (Model 2d in  Figure~\ref{fig7}).

In model 3 of series 6 experiments with thickest intermediate
d\'{e}collement the deformation started by formation of rootless minor
fold. As deformation continued in the next step of shortening, the
low-amplitude lift-off d\'{e}collement fold with extensional
fracture/faults in the core of anticline appeared in the 20\% of bulk
shortening that evolved by uplift and formation of disharmonic minor
folds (Model 3c in  Figure~\ref{fig7}). This fold is isoclinally folded
in the core of the fold that named box lift-off (Model 3c in 
Figure~\ref{fig7}). During progressive deformation, shortening was
accommodated by uplift, decoupling, folding with an increase in limb
dips at the final stage of the experiment. In this stage all minor and
major folds are tightened and the chevron shaped lift-off fold formed
(Model 3d in  Figure~\ref{fig7}). Total displacement magnitude and
direction between the driving hinterland wall and the foreland wall are
illustrated and colored for all models (Models 1e, 2e and 3e in 
Figure~\ref{fig7}) as the characteristic of the tectonic shortening
accommodated by internal deformation.

\section{Comparison with a natural example: the Zagros fold-and-thrust
belt of Iran}\label{sec4}
D\'{e}collements are necessary to form thin-skinned fold-and-thrust
belts and their role has been studied for a long time 
\citep[e.g.,][]{Wilkersonetal2007,Lacombeetal2016,Humairetal2020}. The
DE-model results obtained in this study demonstrate that the presence,
thickness, mechanical properties, and the number of d\'{e}collements
play key roles in controlling how contractional settings evolve. In
this regard, styles of natural fold-and-thrust belts are compared to
our modelling results to illustrate the geometry and kinematics of
their evolution.

The Zagros orogenic belt, SW Iran, resulted from convergence and
shortening between the Arabian plate and the Iranian continental block 
\citep{Alavi2004,Alavi2007,Mouthereauetal2012}. The Zagros orogenic
belt can be divided into several structural zones along strike from the
east to the west including the Fars, 
Izeh, Dezful Embayment, and
Lurestan, based on their differences in structural style and
\mbox{tectono-sedimentary}\ 
\citep[e.g.,][]{Alavi2004,Alavi2007,Mouthereauetal2012}. The Zagros
fold-and-thrust belt is characterized by a broad symmetric to
asymmetric, and gently to isoclinal fold geometries 
\citep{Sherkatietal2005,Sherkatietal2006,Derikvandetal2018,NabaviFossen2021,Shamszadehetal2022b}. The dominant wavelength of such fold structures is 
$15.8\pm5.3$~km \citep{Mouthereauetal2006}. The Zagros fold-and-thrust
belt contains several evaporate and/or weak shale layers in the
sedimentary sequence which are thick enough to form independent
d\'{e}collements. The nature and influence of the presence of multiple
d\'{e}collements are identified by many researches as an important
controlling factor of the structural style in the Zagros belt
\citep[e.g.,][]{Sherkatietal2005,Sherkatietal2006,Sepehretal2004,
Vergesetal2011,Farzipour-SaeinKoyi2014,Ghanadianetal2017a,
Ghanadianetal2017b,Derikvandetal2018,Derikvandetal2019,
Heydarzadehetal2020,Heydarzadehetal2021,KoyiMansurbeg2021}. 
The Dezful Embayment in the Zagros foreland is a classic example of
this case, where containing the world's largest hydrocarbon reserves
in a fold belt. In the fold belt, as many as four Jurassic--Cretaceous
shale units provide intermediate d\'{e}collements above the deep
Infracambrian Hormoz salt d\'{e}collement.

\begin{figure*}
\includegraphics{fig08}
\caption{\label{fig8}Structural cross-section on the southern part of
the Izeh zone and the northern part of the Dezful Embayment. The Hormoz
series and the Garau to Dashtak formations controlled the geometry of
folds as the major d\'{e}collement horizons \citep{Derikvandetal2018}.}
\vspace*{-2pt}
\end{figure*}

Salt/shale-cored d\'{e}collements folds and some isoclinal lift-off
fold geometries are similar to those commonly reported from the other
thin-skinned fold-and-thrust belts, however, thrust-related folds and
the emplacement of thrust sheets have taken place here due to the
presence of multiple d\'{e}collement horizons  \citep{Plaetal2019}.
Evaporite-cored anticlinal structures formed as d\'{e}collement folds
during tectonic shortening and the ductile flow of evaporite in the
anticline core are resulting in an evaporite thickening. Subsequently,
deformation is localized in the evaporite-cored anticlinal structure,
which are structurally accommodated by fore-thrusts and back-thrusts
generating fault-propagation folds and pop-up structures. Fishtail
structures, which are a type of triangle zones in thrust systems, are
also one of the other common structures in contractional deformation
zones with multiple d\'{e}collements. In this area, short wavelength
anticlines in the Oligo-Miocene Asmari Formation are interpreted to be
detached in the Pabdeh and Gurpi or in the Kazhdumi formations 
\citep{Sherkatietal2005,Derikvandetal2018,Derikvandetal2019,Heydarzadehetal2021,Shamszadehetal2022b}. 
Above these shale d\'{e}collements, the Miocene Gachsaran Formation
provides a shallow salt d\'{e}collement 1 to 2 km thick in the Dezful
Embayment 
\citep{Sherkatietal2006,Ghanadianetal2017a,Ghanadianetal2017b,Derikvandetal2018,Derikvandetal2019}. 
Structural coupling across this shallow d\'{e}collement is high, so
that anticlines above the d\'{e}collement tend to overlie anticlines
below the d\'{e}collement. In contrast, where structural coupling is
low, folds form disharmonic geometries above and below the Gachsaran
d\'{e}collement, and create exploration challenges in the Asmari
Limestone reservoir.

Furthermore, the Gachsaran evaporates may migrate from anticlines to
adjacent synclines in the Lurestan structural zone, in which thrusts
affecting the overlying sequences and develop significant disharmonic
fold geometries. Such disharmonic fold geometries in the overlying
Asmari Formation occurs where the thickness of the interposed
incompetent units exceeds a critical thickness that leaves the Asmari
limestone outside of the contact strain zone of the Bangestan
limestones buckles. Disharmonic fold geometries, however, vanishes
toward the southwest, where the Asmari limestone and the top of the
Late Cretaceous Bangestan limestones are roughly concordant, implying a
change in the d\'{e}collement properties. Sedimentary package of the
Oligocene--Miocene Shahbazan--Asmari unit and the Bangestan Group was
folded harmonically and display short-wavelength in geometry 
\citep{Cascielloetal2009,Vergesetal2011,Farzipour-Saein2017}.
Disharmonic\unskip\break 
d\'{e}collement folds can be seen in DE-models of series 4
and also DE-models 2 and 3 of series 6 experiments in the present study
that are buckle folds on different wavelength. Disharmonic fold
geometries across d\'{e}collements may produce sharp transition of
geometric characteristics due to the interlayering of competent and
incompetent beds.  In addition, it should be noted that several
parallel or concentric folding structures have been developed in Zagros
fold-and-thrust belt. Such parallel folds exhibit broad anticlines and
narrow synclines in the upper units and vice versa in the intermediate
and upper d\'{e}collements, that is particularly comparable with
DE-models of series 4 experiments in the present study (i.e. Model 3c
and 3d in  Figure~\ref{fig5}). Furthermore, the lithostratigraphic
units between two major d\'{e}collements in the Zeloi anticline (Dezful
Embayment, SW Iran) display different fold geometries 
(Figure~\ref{fig8})  \citep{Derikvandetal2019}, which is comparable
with the model 3 of series 6 that has thicker intermediate
d\'{e}collement compared to the other two models. Overall, models with
thick d\'{e}collements show a further developed deformation and more
complex fold and fault structures, partically in the overlying strata,
at the final stage of shortening (i.e. 30\%) compared to models with
thin d\'{e}collements (like the model 2 of series 6).\looseness=-1

\section{Comparison to previous modelling studies}\label{sec5}
Critical tapered-wedge theory has been successfully applied to the
study of the deformation mechanism and kinematic 
evolution of 
fold-and-thrust belts and accretionary prisms in contractional 
\mbox{settings}\ 
\citep[e.g.,][]{Davisetal1983,Dahlen1984,Dahlen2003,Buiter2012}. The
theory indicates that the critically tapered wedge, as a stable
geometry, slides along the basal d\'{e}collement without internal
deformation and is dependent on the basal friction coefficient and the
internal strength of the material. According to the modelling results,
when the wedges are in subcritical or supercritical states, the
ramp--flat structure and vertical uplift in the hinterland increases the
slope angle to approach the critical angle. Furthermore,
hinterland-verging fault-propagation folds, as a backthrust, were
developed in DE-models when the wedge at the front of the deformation
belt is in a subcritical state, backthrusting along the intermediate
and/or upper d\'{e}collement steepens the wedge so as to approach the
critical angle.

Many thin-skinned fold-and-thrust belts are formed above multiple
d\'{e}collements producing different structural styles with complicated
fold and fault geometries. So that, fold geometry in fold-and-thrust
belts is an important issue for hydrocarbon exploration 
\citep{NabaviFossen2021}. In thin-skinned fold-and-thrust belts,
d\'{e}collements are frequent, and their impact has long been explored
by means of various techniques 
\citep{DavisEngelder1985,ThomasWesley1995,KonstantinovskayaMalavieille2011,Zhidongetal2021}.
In this regard, the structural evolution of thin-skinned
fold-and-thrust belts has been investigated in several analogue and
numerical models. Overall, there is a general agreement between our
numerical modelling results with scaled analogue and numerical models
that simulating the evolution of multiple d\'{e}collement thin-skinned
fold-and-thrust belts. 

\citet{Ghanadianetal2017b}  (Figure~\ref{fig9}) have documented how an
additional intermediate d\'{e}collement and its thickness variations
promotes folding and thrusting of the heterogeneous sedimentary
sequence in the Dezful Embayment, as well as the role of strain rate.
We numerically obtained a similar evolution with two d\'{e}collements
that the upper d\'{e}collement surrounded by competent layers (i.e.
series 1 experiments in this study) and three d\'{e}collements in which
the intermediate d\'{e}collement thickness varies (i.e. series 6
experiments in this study)  (Figure~\ref{fig7}). 
\citet{Farzipour-Saein2017} and  \citet{Farzipour-SaeinKoyi2014} also
investigated the structural effects of basal and intermediate
d\'{e}collements in the Lurestan, Fars and Izeh structural zones using
a series of scaled sand-box analogue models. Their results and our
numerical models with two weak d\'{e}collements are comparable.

\begin{figure*}
\includegraphics{fig09}
\caption{\label{fig9}(a)~Model 4 from the study conducted 
\citet{Ghanadianetal2017a,Ghanadianetal2017b}
is compared to our modelling outcome (b) and (c). that are
series 1 model 3d and series 6 model 2d from our numerical
investigation respectively.}
\end{figure*}

Furthermore, the analogue modelling results by  \citet{Plaetal2019}
demonstrate that as the viscosity of the d\'{e}collement increases, the
deformation propagates more slowly from hinterland to the foreland part
of the fold-and-thrust belt, and the underlying thrust stack becomes
broader and lower and has a gentler thrust taper angle. The rheology of
the d\'{e}collement defines the distribution and geometry of the
sedimentary cover detached on it that in turn influence the structural
style and development of fold and thrust structures.

Moreover, \citet{Cuietal2020} investigated the role of a strong basal
d\'{e}collement and a weak upper d\'{e}collement with different
strengths by a series of analogue models to simulate the structural
style and deformation patterns of the Longmen Shan fold-and-thrust
belt, eastern margin of the Tibetan Plateau. Their modelling results
show that models with no upper d\'{e}collement and/or with a strong
frictional upper d\'{e}collement deform in the in-sequence
forward-breaking style. However, models with a strong basal
d\'{e}collement and weak upper d\'{e}collement deform in the
out-of-sequence style. Hence, the spatial relation of a strong basal
and weak upper d\'{e}collements may be one of the important factors
producing the structural style.

Furthermore, there are several numerical models that have been applied
to explain the tectonic evolution of thin-skinned fold-and-thrust
belts.  \citet{Morgan2015} and  \citet{Hughes2020} highlighted the
essential role of cohesive strength and residual strength of the basal
d\'{e}collement in the nucleation of discrete faults, the specific
styles of structures that develop in fold-and-thrust belts using a
series of 2D discrete element models. Also, 
\citet{MengHodgetts2019a,MengHodgetts2019b} used 2D discrete element
method to simulate thin-skinned tectonic deformation in stratigraphic
sequences with two d\'{e}collements, to yield new insights into the
combined control of rock competence, depth and thickness of the upper
d\'{e}collement on structural styles and decoupling characteristics. In
this regard, rock rheology and d\'{e}collement layer thickness can
greatly influence thin-skinned tectonics in systems with multiple
d\'{e}collements, and determine the degree of structural decoupling.
Their DE-modelling results indicate models with a lower cohesion and a
thinner d\'{e}collement resulted in more distributed strain and a
larger number of folds, whilst models with a higher cohesion and a
thicker d\'{e}collement led to localized shear zones in fewer folds. In
addition, surface uplift and fold amplitude are mainly positively
affected by the d\'{e}collement thickness, so that, the greater the
surface uplift and fold amplitude are developed in models with the
thicker d\'{e}collement  \citep{MengHodgetts2019a,MengHodgetts2019b}
(such as  Figure~\ref{fig10}). We numerically obtained comparable
results and similar evolution in model series 1 to 3 of the present
study with basal and upper d\'{e}collements within the cover sequence,
which are structurally comparable with DE-models by 
\citet{MengHodgetts2019a,MengHodgetts2019b}. Modelling results of the
present study indicate that competent layers are characterized by
localized strain zones as fault-bend, fault-propagation folds and
pop-up structures with tensile fractures developed in fold hinges.
Moreover, thicker intermediate and upper d\'{e}collements can provide
the higher the degree of decoupling, sufficient mobile materials to
fill fold cores and development of disharmonic folds.

\begin{figure*}
\includegraphics{fig10}
\caption{\label{fig10}DE-modelling results of model 5 in series 2 with
a thin upper d\'{e}collement; and model 14 in series 6 with a thick
upper d\'{e}collement  \citep[after][]{MengHodgetts2019b}.}
\vspace*{3pt}
\end{figure*}

\section{Conclusions}\label{sec6}
Thin-skinned fold--thrust belts commonly contain multiple
d\'{e}collements that exhibit first-order control on their evolution
and structural styles. D\'{e}collements cause the fold style and
fold--fault interactions to become more complex, 
as well as structures
formed under the influence of d\'{e}collements are the locations where
hydrocarbons can be found. Another known factor is the importance of
mechanical property contrast between layers based on strength
differences. As a result, numerical modelling is used as one of the
methods available to better investigate the effect of the above
variables. Therefore, in this paper, we focused on the role of a few
core factors to study of the structural evolution of thin-skinned
fold-and-thrust belts. We have used the discrete element method (DEM)
to carry out a series of numerical models to analyze the differences in
the structural evolution and style of thin-skinned fold-and-thrust
belts causing from the thickness of d\'{e}collements, the number of
d\'{e}collements, the rheological contrast. Finally, we tried to
compare the resulting models to structures and kinematic features of
natural structures, however we do not claim that these models exactly
reflect natural structures.

As a result of horizontal shortening, a variety of fold styles, such as
d\'{e}collement, fold-bend and fold-propagation folds were generated by
varying d\'{e}collement thickness and cover rock cohesiveness. Thicker
d\'{e}collements can develop greater surface uplifts and provide
sufficient materials to fill fold cores and development of disharmonic
and larger folds in geometry. The presence of a thick intermediate
d\'{e}collement facilitated decoupling on cover rock, forming a
significant contrast in the deformation pattern between the underlying
and overlying layers. The modelling result suggests that the presence
of d\'{e}collements and their number and mechanical properties are
considered to be a key parameter in the evolution, structural style,
deformation pattern and strain partitioning of thin-skinned
fold-and-thrust belts. The modelling results are compatible with the
Zagros fold-and-thrust belt, SW Iran, especially the Dezful Embayment
zone, as well as previous analogue and numerical models, regarding the
structural styles, indicating that the combination of increasing
thickness in intermediate and upper d\'{e}collements and a
less-competent cover rock could preferentially produce a range of fold
styles including sinusoidal, box, fault-propagation and fault-bend
folds. This study demonstrates a strong influence of d\'{e}collement
thickness and rock mechanical properties on the structural styles of
fold-and-thrust belts.

{\vspace*{-1pt}}

\section*{Conflicts of interest}
Authors have no conflict of interest to declare.

{\vspace*{-1pt}}

\section*{Author contributions}
NE: Project administration, Field work, Conceptualization, Methodology,
Software, Formal analysis, Validation, Investigation, Visualization,
Resources, Data Curation, Writing---original draft, review and editing.
SAA: Supervision, Investigation, Resources, Data Curation. STN:
Conceptualization, Conceptualization, Investigation, Visualization,
Resources, Data Curation, Writing---review and editing. MRG: Resources,
Writing---review and editing.

{\vspace*{-1pt}}

\section*{Acknowledgements}
The first author greatly thanks  Professor Martin Sch\"{o}pfer
(University of Wien, Austria) for the excellent guidelines and feedback
on an early version of modelling procedures.

{\vspace*{-1pt}}

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